Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 12 \sqrt{2} \sqrt{y} \cos{\left(x^{3} \right)} + 3 y^{3} \sin{\left(x^{3} \right)}=-15

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 36 \sqrt{2} x^{2} \sqrt{y} \sin{\left(x^{3} \right)} + 9 x^{2} y^{3} \cos{\left(x^{3} \right)} + 9 y^{2} y' \sin{\left(x^{3} \right)} + \frac{6 \sqrt{2} y' \cos{\left(x^{3} \right)}}{\sqrt{y}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{3 x^{2} \left(- y^{\frac{7}{2}} \cos{\left(x^{3} \right)} + 4 \sqrt{2} y \sin{\left(x^{3} \right)}\right)}{3 y^{\frac{5}{2}} \sin{\left(x^{3} \right)} + 2 \sqrt{2} \cos{\left(x^{3} \right)}}